Nonlinear (Δν=±1,±2…)
8.4 Methods Experimental SetupExperimental Setup
The broadband 2D THz-THz-Raman experiment is shown in Fig. 8.7, and is mostly unchanged from Chapter 7. Two THz pulses are generated from the signal and idler output of an ultrafast optical parametric amplifier (OPA) incident on two DSTMS THz emitters. The THz pulse generated by the signal is vertically polarized, while the idler THz pulse is horizontally polarized. A THz wiregrid polarizer is used to combine the two THz beams in a collinear geometry, which improves the beam overlap and co-propagation of the two THz pulses. Residual near-IR light in the THz path is blocked with a roughened TOPAS plate. The THz pulses are sent through a 7.5:1 Gaussian telescope, and then focused on the sample. The THz field strength at the sample is ∼300 kV/cm [148]. The Raman probe pulse (∼1 µJ, 38 fs, vertically polarized) is generated from the same laser system and focused on the
1.0 mm Ti:S
1 kHz 3.6 mJ 38 fs
Ti:S 80 MHz
5 nJ
Probe (t2)
λ/4 45 fs OPA
500 μJ/
330 μJ Sample
= 1450 nm (Signal)
= 800 nm
= THz
Pol
= 1780 nm (Idler) Pump (t1)
Chop 166.6 Hz
Chop
250 Hz Lock-in
83.3 Hz DSTMS THz Pol
Liquid Sample
THz + 800 nm
Diamond
<100>
Suprasil Quartz
2.0 mm
0.3 mm
(a) (b)
Figure 8.7: (a) A diagram of the broadband 2D-TTR experiment. A THz wiregrid polarizer (labeled ‘THz Pol’) is used to combine the THz beams. (b) Details of the sample cuvette used in the instrument. The front facing diamond window allows broadband THz transmission to the liquid sample. Both the diamond and Suprasil quartz are transparent to the 800 nm Raman probe pulse.
sample collinear to the THz beams. Heterodyne detection of the Raman probe pulse increases the signal-to-noise ratio and allows for phase-sensitive detection of the 2D-TTR response.
Liquid samples are held in a Suprasil quartz cuvette with a front facing diamond window (Fig. 8.7b). The diamond window allows broadband THz transmission to the liquid sample, while the 800 nm Raman probe passes freely through the diamond window, liquid, and back Suprasil quartz window.
Data Analysis
The raw 2D-TTR data from liquid bromoform are shown in Fig. 8.8a. The ori- entational response is detrended out with a single exponential fit, as demonstrated in Chapter 7. This isolates the vibrational coherences on the t1 and t2 axes (Fig.
8.8b). The rephasing and non-rephasing contributions to the signal are extracted with a 2D complex FFT of the total signal. The rephasing contribution is obtained by setting values in the first and third quadrant of the complex FFT to zero and then applying an inverse FFT back to the time domain (Fig 8.8c). Likewise, the non-rephasing contribution is generated by setting the second and fourth quadrant to zero and applying an inverse FFT back to the time domain (Fig. 8.8).
Intensity (Arb. Units) Intensity (Arb. Units) A
B C D
Figure 8.8: (a) The 2D-TTR response of bromoform. (b) Detrending the orienta- tional response and shifting to larger t2values isolates the vibrational coherences.
(c) The rephasing portion of the vibrational coherences. (d) The non-rephasing portion of the vibrational coherences.
RDM Fit Details
The simulation fits of the 2D-TTR spectrum were carried out with the third order perturbative response function and a reduced density matrix (RDM) model, based on the work in Chapter 7. The fitness of each simulated spectrum was computed with the square root of the L1-norm:
F = sÕ
i
(Ei−Si)2, (8.6)
where F is the fitness function, E is the experimental spectrum, S is the simulated spectrum, and the summation is performed over all points in the 2D spectrum. Each simulated and experimental spectrum was normalized to a maximum signal of 1.0 arbitrary units before this calculation. An L2-norm optimization was also tested, and it yielded nearly identical results. Regularization was also applied to the fitness
|𝟏𝟏𝟏𝟏𝟏𝟏⟩
|𝟏𝟏𝟏𝟏𝟏𝟏⟩
Energy/h (THz)
5 10 15
0
|𝟏𝟏𝟏𝟏𝟏𝟏⟩
|𝟏𝟏𝟏𝟏𝟏𝟏⟩
|𝟏𝟏𝟏𝟏𝟎𝟎⟩
label: |𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝟑𝟑⟩
|𝟏𝟏𝟏𝟏𝟏𝟏⟩
|𝟏𝟏𝟏𝟏𝟏𝟏⟩
|𝟎𝟎𝟏𝟏𝟏𝟏⟩ |𝟏𝟏𝟏𝟏𝟏𝟏⟩ |𝟏𝟏𝟎𝟎𝟏𝟏⟩
|𝟑𝟑𝟏𝟏𝟏𝟏⟩ |𝟎𝟎𝟏𝟏𝟏𝟏⟩ |𝟏𝟏𝟎𝟎𝟏𝟏⟩ |𝟏𝟏𝟑𝟑𝟏𝟏⟩
=C
ν6=C
ν3= 2C
ν6= 2C
ν3= 3C
ν6Fixed Parameters
C = μ or α
Figure 8.9: A schematic of the fixed parameters in the RDM fits.
function to prevent overfitting of the data:
F0=F +α|K|, (8.7)
where F0is the regularized fitness function,α is the regularization parameter and K is a vector containing all of the fit parameters. The value of α was maxi- mized to penalize large fit values, without compromising the agreement between the simulated and experimental spectra. Initially, the parameter space was explored with the perturbative response function derived in Chapter 7 (Equation 7.23) and a basin-hopping algorithm [179] with a quasi-newton SLSQP local minimizer [180].
Then the couplings were further optimized with RDM simulations and the SLSQP quasi-newton minimizer [180]. RDM simulations were carried out with the same
methodology as in Chapter 7, except with a pulse duration of 150 fs and center frequency of 2.5 THz for both THz pulses.
To reduce the parameter space of the minimization, the 1-quantum couplings were fixed at values expected of a harmonic oscillator for dipole transitions: µn−1,n =
√nµ0,1 and Raman transitions: αn−1,n = √
nα0,1. Here, µ0,1 and α0,1 are the THz and Raman coupling elements of the fundamental transitions, respectively. A full schematic of the fixed 1-quantum parameters is given in Fig. 8.9.
The relative intensities of the transition dipole moments of the ν3 (µν3) and ν6 (µν6) fundamentals were calculated using the ratio of the integrated absorbances of the two modes in the linear spectrum. From our previous linear data (Fig.
7.5, the ratio of the integrated absorbance isAν3/Aν6=1.1(1), in agreement with the literature values [187]Aν3/Aν6=1.2. Using our value ofAν3/Aν6=1.1 and assuming a temperature=295 K, the dipole coupling ratio is calculated from
Aν3 Aν6 =
Íν3|n1/2µν3|2∆Nn
Íν6|m1/2µν6|2∆Nm
= |µν3|2Í
ν3n∆Nn
|µν6|2Í
ν6m∆Nm
. (8.8)
The sums are performed over all of the thermally populated 1-quantum transitions (including hot bands) ofν3 andν6. We truncate this sum at 15 THz of total energy.
Thenindices indicate the number ofν3 quanta in the upper state and themindices the number ofν6 quanta in the upper state. For each transition,∆N is the difference in population between the two states involved in each transition. We have also assumed the harmonic oscillator approximation, with the relative transition dipoles of the hot bands given by m1/2µν6 for ν6 and n1/2µν3 for ν3, where µν6 and µν3 are the fundamental transition dipole moments of the two modes. Rearranging this equation we find
µν3 µν6 =
Aν3Í
ν6m∆Nm
Aν6Í
ν3n∆Nn
1/2
≈ 1.6. (8.9)
For the polarizability couplings, theν6 (αν6) fundamental was fixed and the relative intensity ofν3 (αν3), R, was a free parameter in the fit. In this case,αν3is expected to be smaller than αν6, since ν6 is depolarized, ν3 is polarized, and the Raman probe detection is depolarized. Couplings larger than 9.5 THz were set to zero. All other multi-quantum polarizability and dipole couplings were initialized to random values between 0.0-1.0 arbitrary units and constrained to this same range in the fit.
The basin-hopping minimization was run 10 times with the random initialization to
States Quanta HF MP2 CCSD Fits
|0 0 0i, |1 0 0i 1 1.00 1.00 1.00 1.00 (fixed)
|0 0 0i, |0 0 1i 1 1.34 1.00 1.01 1.60 (fixed)
|1 0 0i, |0 0 1i 2 0.04 0.03 0.04 0.01
|2 0 0i, |1 0 1i 2 0.06 0.04 0.06 0.00
|1 0 1i, |0 0 2i 2 0.06 0.04 0.06 0.00
|1 0 0i, |0 0 2i 3 0.00 0.00 0.00 0.22
|0 0 1i, |2 0 0i 3 0.01 0.02 0.01 0.43
|1 0 1i, |3 0 0i 3 0.01 0.03 0.02 0.12
|0 0 1i, |3 0 0i 4 - - - 0.04
|2 0 0i, |0 0 2i 4 - - - 0.22
Table 8.1: Calculated and fit dipole matrix elements relative to µ|000i,|100i. Calcu- lations were performed by Dr. Ralph Welsch. The basis set employed is always aug-cc-pVTZ and all results are for isolated bromoform monomers. In the experi- mental fits, all couplings that connect states of the same energy and change in quanta were constrained to the same value (e.g. µ|001i,|200i, µ|001i,|110i, µ|001i,|020i).
States Quanta HF MP2 CCSD (num.) Fits
|0 0 0i, |1 0 0i 1 1.00 1.00 1.00 1.00 (fixed)
|0 0 0i, |0 0 1i 1 0.92 0.80 0.84 0.06
|1 0 0i, |0 0 1i 2 0.02 0.02 0.02 0.00
|2 0 0i, |1 0 1i 2 0.03 0.03 0.03 0.00
|1 0 1i, |0 0 2i 2 0.03 0.03 0.03 0.00
|1 0 0i, |0 0 2i 3 0.00 0.00 0.03 0.00
|0 0 1i, |2 0 0i 3 0.00 0.00 0.01 0.03
|1 0 1i, |3 0 0i 3 0.00 0.00 0.01 0.00
|0 0 1i, |3 0 0i 4 - - - 0.00
|2 0 0i, |0 0 2i 4 - - - 0.00
Table 8.2: Calculated and fit polarizability matrix elements relative toα|000i,|100i. Calculations performed by Dr. Ralph Welsch. The basis set employed is always aug-cc-pVTZ and all results are for isolated bromoform monomers. CCSD polariz- abilities are obtained by numerical differentiating the dipole moments with respect to an external electric field. In the experimental fits, all couplings that connect states of the same energy and change in quanta were constrained to the same value (e.g.
α|001i,|200i,α|001i,|110i,α|001i,|020i).
check the robustness of the fit. A ‘temperature’ parameter of 0.1 and 10 total basin hopping iterations were used in each run. The final coupling fit is shown in Tables 8.1, 8.2. To confirm that the fit had converged, we calculated the Hessian of the fitness function and verified that there were no negative eigenvalues.
The eigenstate energies were initialized from our previous linear spectrum with
State E (THz) g
|00i 0.0 1
|10i 4.7 2
|01i 6.6 1
|20i 9.4 3
|11i 11.3 2
|02i 13.2 1
|30i 14.1 4
Table 8.3: Eigenstate energies (E), and degeneracies (g) used in the calculations.
ν6=4.76 and ν3=6.68 and manually varied by the experimental error of±0.1 THz.
All combination, overtone, and difference band transitions were determined assum- ing zero vibrational anharmonicity (equal spacing between eigenstates in a particular manifold). The resulting best fit is shown in Table 8.3.